While trying to prove one property of commutative rings with units I can't prove one fact without assuming existence of infinitely many different prime ideals or elements. I tried to test if it was the neccesary assumptions, but I failed, since I don't know any "toy"-examples of such rings.

I know only one example of this kind ($\mathbf{Q} [x]/ (x^2) $ ) but it's not a domain.

So,

Are there any infinite domains with finite number of prime ideals?

If no, then are there any infinite domains with finite (but nontrivial) number of prime elements?

I am interested in noncommutative examples as well.
Sorry if this question is too elementary.

For a noncommutative example you could take the first Weyl algebra $A_1(k)$ with generated over a field $k$ by elements $x$ and $y$ subject to the relation $xy-yx-1$. This is a Noetherian domain which is simple, that is the only two-sided ideals are 0 and the ring itself.